Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 9

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + exp(-x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + exp(-x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\cosh x}^{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (cosh x) -1.0))

C

double code(double x) {
	return pow(cosh(x), -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cosh(x) ** (-1.0d0)
end function

Java

public static double code(double x) {
	return Math.pow(Math.cosh(x), -1.0);
}

Python

def code(x):
	return math.pow(math.cosh(x), -1.0)

Julia

function code(x)
	return cosh(x) ^ -1.0
end

MATLAB

function tmp = code(x)
	tmp = cosh(x) ^ -1.0;
end

Wolfram

code[x_] := N[Power[N[Cosh[x], $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{e^{x} + e^{-x}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x} + e^{-x}}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x}} + e^{-x}} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{2}{e^{x} + \color{blue}{e^{-x}}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{2}{e^{x} + e^{\color{blue}{\mathsf{neg}\left(x\right)}}} \]

   6. cosh-undef N/A

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \cosh x}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\cosh x} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   10. lower-cosh.f64 100.0

       \[\leadsto \frac{1}{\color{blue}{\cosh x}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Final simplification 100.0%

   \[\leadsto {\cosh x}^{-1} \]
6. Add Preprocessing

Alternative 2: 91.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (fma
   (*
    (fma (* (fma (* x x) 0.001388888888888889 0.041666666666666664) x) x 0.5)
    x)
   x
   1.0)
  -1.0))

C

double code(double x) {
	return pow(fma((fma((fma((x * x), 0.001388888888888889, 0.041666666666666664) * x), x, 0.5) * x), x, 1.0), -1.0);
}

Julia

function code(x)
	return fma(Float64(fma(Float64(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664) * x), x, 0.5) * x), x, 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{e^{x} + e^{-x}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x} + e^{-x}}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x}} + e^{-x}} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{2}{e^{x} + \color{blue}{e^{-x}}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{2}{e^{x} + e^{\color{blue}{\mathsf{neg}\left(x\right)}}} \]

   6. cosh-undef N/A

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \cosh x}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\cosh x} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   10. lower-cosh.f64 100.0

       \[\leadsto \frac{1}{\color{blue}{\cosh x}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   14. lower-*.f64 91.5

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 91.5%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 91.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, \color{blue}{x}, 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 91.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right) \cdot x, x, 1\right)} \]

12. Final simplification 91.5%

    \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \]
13. Add Preprocessing

Alternative 3: 91.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (fma (* (fma (* (* x x) 0.001388888888888889) (* x x) 0.5) x) x 1.0)
  -1.0))

C

double code(double x) {
	return pow(fma((fma(((x * x) * 0.001388888888888889), (x * x), 0.5) * x), x, 1.0), -1.0);
}

Julia

function code(x)
	return fma(Float64(fma(Float64(Float64(x * x) * 0.001388888888888889), Float64(x * x), 0.5) * x), x, 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{e^{x} + e^{-x}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x} + e^{-x}}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x}} + e^{-x}} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{2}{e^{x} + \color{blue}{e^{-x}}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{2}{e^{x} + e^{\color{blue}{\mathsf{neg}\left(x\right)}}} \]

   6. cosh-undef N/A

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \cosh x}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\cosh x} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   10. lower-cosh.f64 100.0

       \[\leadsto \frac{1}{\color{blue}{\cosh x}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   14. lower-*.f64 91.5

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 91.5%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 91.3%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \]
11. Step-by-step derivation

12. Applied rewrites 91.3%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right) \cdot x, \color{blue}{x}, 1\right)} \]

13. Final simplification 91.3%

    \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \]
14. Add Preprocessing

Alternative 4: 91.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (fma
   (* (* (fma 0.001388888888888889 (* x x) 0.041666666666666664) x) x)
   (* x x)
   1.0)
  -1.0))

C

double code(double x) {
	return pow(fma(((fma(0.001388888888888889, (x * x), 0.041666666666666664) * x) * x), (x * x), 1.0), -1.0);
}

Julia

function code(x)
	return fma(Float64(Float64(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664) * x) * x), Float64(x * x), 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{e^{x} + e^{-x}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x} + e^{-x}}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x}} + e^{-x}} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{2}{e^{x} + \color{blue}{e^{-x}}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{2}{e^{x} + e^{\color{blue}{\mathsf{neg}\left(x\right)}}} \]

   6. cosh-undef N/A

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \cosh x}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\cosh x} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   10. lower-cosh.f64 100.0

       \[\leadsto \frac{1}{\color{blue}{\cosh x}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   14. lower-*.f64 91.5

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 91.5%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x} \cdot x, 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 90.9%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, \color{blue}{x} \cdot x, 1\right)} \]

11. Final simplification 90.9%

    \[\leadsto {\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right)\right)}^{-1} \]
12. Add Preprocessing

Alternative 5: 69.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.20833333333333334, x \cdot x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4)
   (fma (fma 0.20833333333333334 (* x x) -0.5) (* x x) 1.0)
   (pow (* (* (fma 0.041666666666666664 (* x x) 0.5) x) x) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = fma(fma(0.20833333333333334, (x * x), -0.5), (x * x), 1.0);
	} else {
		tmp = pow(((fma(0.041666666666666664, (x * x), 0.5) * x) * x), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = fma(fma(0.20833333333333334, Float64(x * x), -0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(fma(0.041666666666666664, Float64(x * x), 0.5) * x) * x) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(N[(0.20833333333333334 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[Power[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}}, {x}^{2}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{5}{24} \cdot {x}^{2}} - \frac{1}{2}, {x}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{5}{24} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{5}{24} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{2}, {x}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{5}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, \color{blue}{x \cdot x}, 1\right) \]

      9. lower-*.f64 70.6

         \[\leadsto \mathsf{fma}\left(0.20833333333333334 \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.20833333333333334 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{x} \cdot x, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.20833333333333334, x \cdot x, -0.5\right), \color{blue}{x} \cdot x, 1\right) \]

   if 1.3999999999999999 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{e^{x} + e^{-x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{e^{x} + e^{-x}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{e^{x}} + e^{-x}} \]

      4. lift-exp.f64 N/A

         \[\leadsto \frac{2}{e^{x} + \color{blue}{e^{-x}}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{2}{e^{x} + e^{\color{blue}{\mathsf{neg}\left(x\right)}}} \]

      6. cosh-undef N/A

         \[\leadsto \frac{2}{\color{blue}{2 \cdot \cosh x}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\cosh x} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

      10. lower-cosh.f64 100.0

          \[\leadsto \frac{1}{\color{blue}{\cosh x}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

      9. lower-*.f64 72.2

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

   7. Applied rewrites 72.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{{x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.2%

       \[\leadsto \frac{1}{\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot \color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.20833333333333334, x \cdot x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) -1.0))

C

double code(double x) {
	return pow(fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0), -1.0);
}

Julia

function code(x)
	return fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{e^{x} + e^{-x}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x} + e^{-x}}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x}} + e^{-x}} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{2}{e^{x} + \color{blue}{e^{-x}}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{2}{e^{x} + e^{\color{blue}{\mathsf{neg}\left(x\right)}}} \]

   6. cosh-undef N/A

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \cosh x}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\cosh x} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   10. lower-cosh.f64 100.0

       \[\leadsto \frac{1}{\color{blue}{\cosh x}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   9. lower-*.f64 88.1

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 88.1%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]

8. Final simplification 88.1%

   \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\right)}^{-1} \]
9. Add Preprocessing

Alternative 7: 87.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (fma (* 0.041666666666666664 (* x x)) (* x x) 1.0) -1.0))

C

double code(double x) {
	return pow(fma((0.041666666666666664 * (x * x)), (x * x), 1.0), -1.0);
}

Julia

function code(x)
	return fma(Float64(0.041666666666666664 * Float64(x * x)), Float64(x * x), 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{e^{x} + e^{-x}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x} + e^{-x}}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{e^{x}} + e^{-x}} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{2}{e^{x} + \color{blue}{e^{-x}}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{2}{e^{x} + e^{\color{blue}{\mathsf{neg}\left(x\right)}}} \]

   6. cosh-undef N/A

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \cosh x}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\cosh x} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   10. lower-cosh.f64 100.0

       \[\leadsto \frac{1}{\color{blue}{\cosh x}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   9. lower-*.f64 88.1

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 88.1%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, \color{blue}{x} \cdot x, 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 87.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), \color{blue}{x} \cdot x, 1\right)} \]

11. Final simplification 87.5%

    \[\leadsto {\left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right)\right)}^{-1} \]
12. Add Preprocessing

Alternative 8: 76.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, x, 2\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (fma x x 2.0)))

C

double code(double x) {
	return 2.0 / fma(x, x, 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, x, 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{{x}^{2} + 2}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{x \cdot x} + 2} \]

   3. lower-fma.f64 76.6

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Applied rewrites 76.6%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]
6. Add Preprocessing

Alternative 9: 51.4% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 52.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))